Problem: Two cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a King and the second card is a $\heartsuit$?
Solution: We have two cases because if the first card is a King, it could be a $\heartsuit$ or not be a $\heartsuit$.

There is a $\dfrac{1}{52}$ chance that the King of $\heartsuit$ is drawn first, and a $\dfrac{12}{51} = \dfrac{4}{17}$ chance that the second card drawn is one of the twelve remaining $\heartsuit$, which gives a probability of $\dfrac{1}{52} \times \dfrac{4}{17} = \dfrac{1}{221}$ chance that this occurs.

There is a $\dfrac{3}{52}$ chance that a non-$\heartsuit$ King is drawn first, and a $\dfrac{13}{51}$ chance that a $\heartsuit$ is drawn second, giving a $\dfrac{3}{52} \times \dfrac{13}{51} = \dfrac{1}{68}$ chance that this occurs.

So the probability that one of these two cases happens is $\dfrac{1}{221} + \dfrac{1}{68} = \boxed{\dfrac{1}{52}}$.